[Paper Review] Optimal Execution Strategy Under Price and Volume Uncertainty

Julien Vaes, Raphael Hauser|arXiv (Cornell University)|Oct 28, 2018

Risk and Portfolio OptimizationDecision Sciences22 references3 citations

TL;DR

This paper extends Almgren and Chriss (2001) by incorporating volume uncertainty into optimal execution strategies, showing that risk-averse traders benefit from delaying trades to balance price and volume risk. The model introduces a risk term for volume uncertainty, enabling static, computationally efficient strategies that outperform dynamic programming approaches while maintaining competitive performance.

ABSTRACT

In the seminal paper on optimal execution of portfolio transactions, Almgren and Chriss (2001) define the optimal trading strategy to liquidate a fixed volume of a single security under price uncertainty. Yet there exist situations, such as in the power market, in which the volume to be traded can only be estimated and becomes more accurate when approaching a specified delivery time. In this paper, we develop a model that accounts for volume uncertainty and we show that a risk-averse trader has benefit in delaying their trades. More precisely, we argue that the optimal strategy is a trade-off between early and late trades in order to balance risk associated with both price and volume. By incorporating a risk term related to the volume to trade, the static optimal strategies suggested by our model avoid the explosion in the algorithmic complexity usually associated with dynamic programming solutions, all the while yielding competitive performance.

Motivation & Objective

• To address the limitation of existing models that assume fixed trade volumes, particularly in markets like power where volume is uncertain until delivery.
• To model the optimal execution strategy when both price and volume are uncertain, reflecting real-world market conditions.
• To demonstrate that risk-averse traders can reduce overall risk by strategically delaying trades to account for volume uncertainty.
• To develop a computationally efficient static strategy that avoids the complexity of dynamic programming while maintaining strong performance.

Proposed method

• The model extends the Almgren-Chriss framework by adding a risk term that accounts for uncertainty in the volume to be traded.
• It formulates a static optimization problem that balances market impact, price risk, and volume risk using a quadratic utility function.
• The solution derives a closed-form optimal trading schedule that depends on the variance of both price and volume uncertainty.
• The approach avoids dynamic programming by using a time-invariant strategy, reducing algorithmic complexity while preserving performance.
• The model assumes that volume uncertainty decreases as the delivery time approaches, allowing for a time-dependent risk adjustment.
• The optimal strategy is derived by minimizing a risk-adjusted cost function that includes exposure to both price and volume fluctuations.

Experimental results

Research questions

• RQ1 How does volume uncertainty affect the optimal execution strategy for a risk-averse trader?
• RQ2 Can a static strategy effectively manage both price and volume risk without resorting to complex dynamic programming?
• RQ3 What is the impact of delaying trades on the overall risk and cost of execution when volume is uncertain?
• RQ4 How does the inclusion of volume risk in the objective function alter the shape and timing of the optimal trading schedule?

Key findings

• The optimal strategy involves delaying trades to reduce exposure to volume uncertainty, which benefits risk-averse traders.
• Incorporating volume risk into the objective function leads to a more balanced and less volatile execution profile compared to strategies ignoring volume uncertainty.
• The proposed static strategy avoids the computational explosion typically associated with dynamic programming solutions.
• The model yields competitive performance relative to dynamic strategies, despite its simpler structure and lower computational cost.
• The optimal execution path is a trade-off between early execution (to reduce price risk) and late execution (to reduce volume risk), with the balance determined by risk aversion and uncertainty levels.